As an engine mount structure for supporting an engine on a vehicle body while attenuating vibration to a certain extent, various types of structures have been used. As one of the types, Japanese Unexamined Patent Application, First Publication No. 2001-105896 proposes a structure for supporting an engine while an anti-vibration device made of a rubber material is interposed between a bracket on a vehicle body which is mounted to the vehicle body and an arm bracket extending from an engine to attenuate vibration as much as possible.